Further study on MacMahon-type sums of divisors

TL;DR

This paper investigates MacMahon-type sums of divisors for specific parameters, demonstrating their connection to quasi-modular forms and revealing new identities such as a sum involving q^{3n} over (1-q^{3n})^2.

Contribution

It extends previous work by analyzing these sums for a range of parameters, showing they are quasi-modular forms, and discovering new identities like the sum for U_2(1,q).

Findings

All functions are in the ring of quasi-modular forms.

Identified a new sum: U_2(1,q) equals sum of q^{3n}/(1-q^{3n})^2.

Extended the analysis to cases a=0, ±1, ±2.

Abstract

This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots(1+aq^{n_t}+q^{2n_t})} $$ when $a$ is one of $0, \pm 1, \pm2$. The idea builds on our previous treatment of the case $a=-2$. It is shown that all these functions lie in the ring of quasi-modular forms. Among the more surprising findings is $$U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.$$